Lump solitons in a higher-order nonlinear equation in 2+1 ... · The model Singular Manifold Method Lumps PHYSICAL REVIEW E93, 062219 (2016) Lump solitons in a higher-order nonlinear

The modelSingular Manifold Method

Lumps

Lump solitons in a higher-order nonlinear equationin 2 + 1 dimensions

P. G. Estevez.E. Diaz. F. Domınguez-Adame. R. Diez. J.M. Cervero

Area de Fısica TeoricaUniversidad de Salamanca

Burgos. October 2016

P. G. Estevez. E. Diaz. F. Domınguez-Adame. R. Diez. J.M. CerveroLump solitons in a higher-order nonlinear equation in 2 + 1 dimensions


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Table of contents

1 The modelTruncated expansions

Lax pairDarboux transformations

2 Lumps



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PHYSICAL REVIEW E 93, 062219 (2016)

Lump solitons in a higher-order nonlinear equation in 2 + 1 dimensions

P. G. Estevez,1,* E. Dıaz,2 F. Domınguez-Adame,2 Jose M. Cervero,1 and E. Diez1

1Departamento de Fısica Fundamental, Universidad de Salamanca, E-37008 Salamanca, Spain2GISC, Departamento de Fısica de Materiales, Universidad Complutense, E-28040 Madrid, Spain(Received 12 February 2016; revised manuscript received 29 April 2016; published 20 June 2016)

We propose and examine an integrable system of nonlinear equations that generalizes the nonlinear Schrodingerequation to 2 + 1 dimensions. This integrable system of equations is a promising starting point to elaborate moreaccurate models in nonlinear optics and molecular systems within the continuum limit. The Lax pair for the systemis derived after applying the singular manifold method. We also present an iterative procedure to construct thesolutions from a seed solution. Solutions with one-, two-, and three-lump solitons are thoroughly discussed.

DOI: 10.1103/PhysRevE.93.062219

I. INTRODUCTION

The cubic nonlinear Schrodinger equation (NLSE) withadditional high-order dispersion terms emerges very often inthe theoretical description of a number of physical problemsin molecular systems, nonlinear optics, and fluid dynamics,to name a few. For instance, the propagation of energyreleased during adenoside triphosphate hydrolysis, throughamide-I vibrations along the hydrogen bonding spine ofthe α-helical proteins, is described by a set of equationswhich, for dipole-dipole interaction, in the lower order of thecontinuum approximation is governed by the NLSE. Someyears ago, in an attempt to extend the Davydov model for theenergy transfer of local vibrational modes in α-helical proteins[1,2], Daniel and Deepmala considered the effects of higher-order molecular excitations [3] that introduce quadrupole-quadrupole coefficients. The result of such a generalizationis a NLSE that includes a fourth-order dispersion term, theso-called Lakshmanan-Porsezian-Daniel equation:

iψt + 12ψxx + |ψ |2ψ + γ

(ψxxxx + 6ψ2

xψ∗ + 4|ψx |2ψ+8ψxx |ψ |2 + 2ψ∗

xxψ2 + 6ψ |ψ |4) = 0, (1)

whose integrability and soliton solutions were studied inRefs. [3] and [4]. More recently, Ankiewicz et al. proposeda further generalization of the NLSE, adding a third-orderdispersion term [5]. The integrability of this extended NLSEfor some values of the parameters of the equation wasconfirmed in Ref. [6], where Lax operators were presented.This integrable version appears in the mentioned references as

iψt + 12ψxx + |ψ |2ψ + γ

(ψxxxx + 6ψ2

xψ∗ + 4|ψx |2ψ+ 8ψxx |ψ |2 + 2ψ∗

xxψ2 + 6ψ |ψ |4)

= iα(ψxxx + 6ψx |ψ |2). (2)

This equation contains many integrable particular cases suchas the standard NLSE (α = γ = 0) [7], the Hirota equation(γ = 0) [8], and the Lakshmanan-Porsezian-Daniel equation(α = 0) [4]. Soliton solutions and rogue wave for this equationcan be found in Refs. [6] and [7] as well as in Refs. [9] and [10].The relevance of third-order dispersion terms in the contextof the self-induced Raman effect have been pointed out by

*Corresponding author: [email protected]

Hesthaven et al. [11]. Moreover, rogue waves in optical fiberscan be mathematically described by the NLSE equation and itsextensions that take into account third-order dispersion [12].

Models discussed so far are defined in 1 + 1 dimensions asthey are aimed at describing the dynamics of the excitationsin a single strand of the protein. These models need to includemore degrees of freedom and more spatial dimensions tocope with the complex helical geometry of the proteins. Tothis aim, there exist different generalizations of the NLSE to2 + 1 dimensions. In particular, we can consider the followingsystem proposed by Calogero in Ref. [13], and then discussedby Zakharov [14], which trivially reduces to the NLSE on theline x = y.

iut + uxy + 2umy = 0, − iwt + wxy + 2wmy = 0,

mx + uw = 0, (3)

where w = u∗. This equation has been studied by differentauthors. A derivation of the Lax pair and Darboux transfor-mations by means of the singular manifold method appears inRef. [15]. The same method was applied in Ref. [16] to deriverational solitons (lumps) of a different generalization of theNLSE to 2 + 1 dimensions. Notice that the second derivativeincludes crossed terms uxy instead of some combination ofuxx and uyy as appears in many genaralizations of NLS. Theycould be easily recovered through the change of variablesx = x + y and y = x − y that yields uxy = uxx − uyy .

Rogue waves in 1 + 1 dimensions [17] as well as lumps in2 + 1 dimensions [16] are rational solutions with nontrivialbehavior. This suggests that rogue waves can appear as areduction of variables in the lump solutions. As it is wellknown, lumps are solutions whose meromorphic structureguarantees their stability [18]. This is the main motivation topropose a modified NLSE in 2 + 1 dimensions similarly to thegeneralization considered in Ref. [15] but including also third-and fourth-order dispersion terms, as in the case of Eq. (2), tobe a good candidate for the continuum limit of different dis-crete models that have been proposed to describe the dynamicsof α-helical proteins. The proposed generalization of the set(3) can be cast as a system of equations in the following form:

iut + uxy + 2umy + iα(uxxx − 6uwux)

+ γ(uxxxx − 8uwuxx − 2u2wxx

− 4uuxwx − 6wu2x + 6u3w2

) = 0, (4a)

2470-0045/2016/93(6)/062219(8) 062219-1 ©2016 American Physical Society



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high order dispersion terms

The cubic nonlinear Schrodinger equation (NLSE) with additionalhigh order dispersion terms emerges very often in the theoreticaldescription of a number of physical problems in molecular systems

dipole-dipole interaction

The propagation of energy released during ATP hidrolysis throughamide-I vibrations along the hydrogen bonding spine of thealpha-helical proteins is described by a set of equations which, fordipole-dipole interaction, in the lower order of continuumapproximation is governed by NLSE

quadrupole-quadrupole terms

Daniel and Deepmala considered the effects of higher ordermolecular excitations that introduce quadrupole-quadrupolecoefficients.



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Lakshmanan-Porsezian-Daniel (1995)

They considered the effects of higher order molecular excitations.The continuum limit yields a generalization of NLSE that includesa fourth-order dispersion term

iψt +1

2ψxx + |ψ|2ψ + γ

(ψxxxx + 6ψ2

xψ∗ + 4|ψx |2ψ

+ 8ψxx |ψ|2 + 2ψ∗xxψ2 + 6ψ|ψ|4

)= 0 ,

Ankiewitz et and Akhmediev (2014)

An integrable model adding a third-order dispersion term.

iψt +1

2ψxx + |ψ|2ψ + γ

(ψxxxx + 6ψ2

xψ∗ + 4|ψx |2ψ

+ 8ψxx |ψ|2 + 2ψ∗xxψ2 + 6ψ|ψ|4

)= iα

(ψxxx + 6ψx |ψ|2

).



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Lakshmanan-Porsezian-Daniel (1995)

They considered the effects of higher order molecular excitations.The continuum limit yields a generalization of NLSE that includesa fourth-order dispersion term

iψt +1

2ψxx + |ψ|2ψ + γ

(ψxxxx + 6ψ2

xψ∗ + 4|ψx |2ψ

+ 8ψxx |ψ|2 + 2ψ∗xxψ2 + 6ψ|ψ|4

)= 0 ,

Ankiewitz et and Akhmediev (2014)

An integrable model adding a third-order dispersion term.

iψt +1

2ψxx + |ψ|2ψ + γ

(ψxxxx + 6ψ2

xψ∗ + 4|ψx |2ψ

+ 8ψxx |ψ|2 + 2ψ∗xxψ2 + 6ψ|ψ|4

)= iα

(ψxxx + 6ψx |ψ|2

).



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Generalizations of NLS to 2+1

Calogero (1975)

iut + uxy + 2umy = 0 ,

−iwt + wxy + 2wmy = 0 ,

mx + uw = 0 ,

Rogue waves and lumps

Rogue waves in 1 + 1 dimensions as well as lumps in 2 + 1dimensions are rational solutions with non trivial behavior. Thissuggests that rogue waves can appear as a reduction of variables inthe lump solutions. . This is the main motivation to propose amodified NLSE in 2 + 1 dimensions but including also third- andfourth-order dispersion terms.



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Our model (2016)

iut + uxy + 2umy + iα (uxxx − 6uwux)

+γ(uxxxx − 8uwuxx − 2u2wxx

−4uuxwx − 6wu2x + 6u3w2) = 0 ,

−iwt + wxy + 2wmy − iα (wxxx − 6uwwx)

+γ(wxxxx − 8uwwxx − 2w2uxx

−4wuxwx − 6uw2x + 6u2w3) = 0 ,

mx + uw = 0 .



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References

Estevez PG. J. Math. Phys. (1999)

Estevez PG. and Prada J. Theor. Math. Phys. (2007)

Estevez PG. Prada J. and Villarroel J. J. Phys. A (2007)

Villarroel J. Prada J. and Estevez PG. Studies in Applied Math.(2009)

Estevez PG. et al Phys. Rev. E (2016)



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References








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Summary

We propose and examine an integrable system of nonlinearequations that generalizes the nonlinear Schrodinger equationto 2 + 1 dimensions.

This integrable system of equations is a promising startingpoint to elaborate more accurate models in nonlinear opticsand molecular systems within the continuum limit.

The Lax pair for the system is derived after applying thesingular manifold method.

We also present an iterative procedure to construct thesolutions from a seed solution.

Solutions with one, two and three lump solitons arethoroughly discussed.



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Singular manifold equationsLax pairDarboux transformations

Painleve expansion

u =∞∑j=0

aj(x , y , t) [φ(x , y , t)]j−1 ,

w =∞∑j=0

bj(x , y , t) [φ(x , y , t)]j−1 ,

m =∞∑j=0

mj(x , y , t) [φ(x , y , t)]j−1 ,

where φ(x , y , t) is an arbitrary function. This means that allsolutions are single-valued around the singularity manifoldφ(x , y , t) = 0.



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Truncated expansion. Singular manifold

There exists a resonance in j = 0. A function g1 is introduced togive account of this arbitrariness

u[1] = u[0] +g1φ1,x

φ1,

w [1] = w [0] +φ1,x

g1φ1,

m[1] = m[0] +φ1,x

φ1,

Solutions obtained through this expansion in the singular manifoldφ1 have been denoted as (u[1],w [1],m[1]). It implies an iterativemethod of construction of solutions where the superindex [0]denotes a seed solution and [1] the iterated one



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Singular manifold equations

Substitution of the truncated expansion leads to threepolynomials in powers of φ1.

By imposing that each coefficient vanishes, we obtain a set ofequations: the singular manifold equations that relates thesingular manifold φ1 with the seed solution (u[0],w [0],m[0]).

The process of obtaining these equations requires sometedious but straightforward calculation that we haveperformed with the aid of the symbolic calculus packageMAPLE.



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Lax pair: The singular manifold equations can be linearized

ψ1,x + u[0]ϕ1 + iλ1ψ1 = 0 ,

ψ1,t = 2λ1ψ1,y + λ1,yψ1 + i(m

[0]y ψ1 − u

[0]y ϕ1

)+ (α− 2λ1γ)F

[ψ1, ϕ1, λ1, u

[0],w [0]]

+ iγG[ψ1, ϕ1, u

[0],w [0]]

ϕ1,x + w [0]ψ1 − iλ1ϕ1 = 0 ,

ϕ1,t = 2λ1ϕ1,y + λ1,yϕ1 − i(m

[0]y ϕ1 − w

[0]y ψ1

)+ (α− 2λ1γ)F

[ϕ1, ψ1, λ1,w

[0], u[0]]− iγG

[ϕ1, ψ1,w

[0], u[0]]

F [ψ, ϕ, λ, u,w ] = 3(uw + λ2)ψx − ψxxx − 3uxϕx ,

G [ψ, ϕ, u,w ] =(

3u2w2 + uxwx − uwxx − wuxx

)ψ + (6uwux − uxxx )ϕ.



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Non-isospectral Lax pair

λt − 2λλy = 0



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Singular manifold and eigenfunctions

φ1,x = ψ1ϕ1,

φ1,t = 2λ1φ1,y + i (ϕ1ψ1,y − ψ1ϕ1,y )

+ (α− 2λ1γ)J [ψ1, ϕ1, λ1] + iγK[ψ1, ϕ1, u

[0],w [0]].

where we have defined

J [ψ, ϕ, λ] = 4ψxϕx + 6λ2ψϕ− ψϕxx − ϕψxx ,

K [ψ, ϕ, u,w ] = ϕψxx − ψϕxx + 3 (ψxϕxx − ϕxψxx ) + 6uw (ψϕx − ϕψx ) .



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These Lax pairs can be considered as nonlinear equations between the fields

and the eigenfunctions

The crucial point here is to consider the Lax pair as a set ofnonlinear equations for the fields and eigenfunctions together

ψ1,2 = ψ2 − ψ1∆1,2

φ1,

ϕ1,2 = ϕ2 − ϕ1∆1,2

φ1,

where (ψi , ϕi ) are eigenfunctions of the Lax pair for (u[0],w [0])with eigenvalue λj (j = 1, 2).

∆1,2 =i

2

ϕ1ψ2 − ϕ2ψ1

λ2 − λ1.



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These Lax pairs can be considered as nonlinear equations between the fields

and the eigenfunctions

The crucial point here is to consider the Lax pair as a set ofnonlinear equations for the fields and eigenfunctions together

ψ1,2 = ψ2 − ψ1∆1,2

φ1,

ϕ1,2 = ϕ2 − ϕ1∆1,2

φ1,

where (ψi , ϕi ) are eigenfunctions of the Lax pair for (u[0],w [0])with eigenvalue λj (j = 1, 2).

∆1,2 =i

2

ϕ1ψ2 − ϕ2ψ1

λ2 − λ1.



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Summarizing

u[2] = u[0] +1

τ1,2

(ψ1

ψ2

)(∆2,2 −∆1,2

−∆1,2 ∆1,1

)(ψ1

ψ2

),

w [2] = w [0] +1

τ1,2

(ϕ1

ϕ2

)(∆2,2 −∆1,2

−∆1,2 ∆1,1

)(ϕ1

ϕ2

),

m[2] = m[0] +(τ1,2)xτ1,2

,

τ1,2 = φ1,2φ1 = φ2φ1 − (∆1,2)2 = det(∆) ,



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Solve the Lax pair in order to obtain the eigenfunctions(φi , ϕi ) for a given seed solution (u[0],w [0]) and spectralparameter λi .

These eigenfunctions allow us to construct the n× n matrix ∆

∆j ,k =i

2

ϕjψk − ϕkψj

λk − λjj , k : 1...n.

The τ functions of order n can be obtained as thedeterminant of the matrix ∆.

τ1,2,··· ,n = φ1φ1,2 · · ·φ1,2,··· ,n = det(∆) ,

The probability density u[n]w [n] = −m[n]x can be easily

obtained through

m[n]x = m

[0]x +

((τ1,2,··· ,n)xτ1,2,··· ,n

)x

.



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Lumps: u[0] = ij0 , w [0] = −ij0, m[0] = j20x − 3γj4

0y ,

where j0 is an arbitrary constant. We can obtain rational solutionsof the Lax pair as

ψ1 = j1 + j0(j2 − j1){x + b1j

20y + 2

[−3α− ij2

0 (b1 + 6γ)]j0t},

ϕ1 = j2 + j0(j2 − j1){x + b1j

20y + 2

[−3α + ij2

0 (b1 + 6γ)]j0t},

ψ2 = j2 + j0(j2 − j1){x + b1j

20y + 2

[−3α− ij2

0 (b1 + 6γ)]j0t},

ϕ2 = j1 + j0(j2 − j1){x + b1j

20y + 2

[−3α + ij2

0 (b1 + 6γ)]j0t},

where λ1 = −λ2 = ij0

and j1, j2 and b1 are three arbitrary constants. This particularchoice of λ2 as the complex conjugate of λ1 has been made inorder to have φ2 as the complex conjugate of φ1. Our goal is toget a real τ -function without zeroes and therefore solutionswithout singularities.



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τ1,2 = A2 + B2 + C 2 , Zt − 2ij0Zy = 0

A = 12αh1j70 t

2{

2δX + 3y(

3α2 − j20 δ2)}

− 2h1j50 δt

[X 2 + j20 y

2(

27α2 − j20 δ2)]

+ 2h2j04t[δX + y

(9α2 − j02

δ2)]

− h1αj0y[j40 y

2(

9α2 − 3j20 δ2)

+ 2]

− 3h2αδj40 y

2 + 3αj0j1j2y + Re[Z ] ,

B = 4h1j60 t

2{X(

9α2 − 3j20 δ2)− yδj20

(27α2 − j20 δ

2)}

− 6h1αj40 t[X 2 + 3j20 y

2(

3α2 − j20 δ2)]

+ 6h2αj30 t(X − 2δj20 y

)+

1

6h1j

20

[2X 3 + 2δj40 y

3(

27α2 − j20 δ2)]

+1

6h1j

20 [y (7δ − 4b1)]

−1

2h2j0

[X 2 + j20 y

2(

9α2 − 3j20 δ2)]

+ j1j2(X − j20 δy) + Im[Z ]

C =h1

4j0

[2j20 (6j20αt − X )2 + 8j80 δ

2t2 + 1]

+h2

2(6j20αt − X ) +

j1j2

j0,

and we have introduced the notationX = x + b1j

20 y ,

h1 = (j1 − j2)2, h2 = j21 − j22 ,

δ = 6γ + b1 ,



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j1 = j2, Z = 0

τ1,2 = (3αj0y)2 +(X − j2

0 δy)2

+ 1j20



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j1 = j2, Z = a1(y + 2ij0t)2 with a1 real

τ =(3j0αy + a1y

2 − 4a1j20 t

2)2

+(X − j2

0yδ + 4a1yj0t)2

+ 14j2

0



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j1 = j2, Z = a1(y + 2ij0t)3 with a1 real

τ =(3j0αy + a1y

3 − 12a1yj20 t

2)2

+(X − j2

0yδ + 6a1y2j0t − 8a1j

30 t

3)2

+ 14j2

0



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j1 = j2, Z = ia1(y + 2ij0t)2 with a1 real



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j1 = j2, Z = ia1(y + 2ij0t)3 with a1 real



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j2 = −j1



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Documents

Lump solitons in a higher-order nonlinear equation in 2+1 ... · The model Singular Manifold Method Lumps PHYSICAL REVIEW E93, 062219 (2016) Lump solitons in a higher-order nonlinear